Let $P \subset \mathbb R^n$ be a Delzant polytope defined by inequalities $\ell_i(x) \geq 0, i=1, \ldots, d$.

Of course, from the symplectic point of view, the inequalities $a_i \ell_i \geq 0$ still define the same polytope for all positive real $a_i$.

What I wonder is the following. Does the function $g: P \rightarrow \mathbb R$ (up to affine terms) $$ g(x) = \sum_{i=1}^d (a_i \ell_i(x) \log \ell_i(x)) + h(x) $$ with $h$ smooth on $P$ define a compatible complex structure on $P$, no matter what the $a_i>0$ are?

Or are legal potentials only the ones where the $\ell_i(x)$ are of the form $\langle x, \mu_i \rangle - \lambda_i$ where $\mu_i$ is a primitive element of the integral lattice and points to the interior of the $i$-th face?

thank you David